Nekrasov Functions and Exact Bohr-Sommerfeld Integrals

TL;DR

This paper explores the connection between Nekrasov functions, Bohr-Sommerfeld integrals, and sine-Gordon models, revealing a new link between Liouville and sine-Gordon theories and proposing a novel quantization perspective.

Contribution

It demonstrates how the Nekrasov prepotential can be derived from sine-Gordon model monodromies, linking it to Liouville theory and suggesting a new approach to quantization.

Findings

Nekrasov prepotential relates to sine-Gordon monodromies.

The epsilon parameters correspond to Planck constant and quantization.

Proposes a new link between Liouville and sine-Gordon theories.

Abstract

In the case of SU(2), associated by the AGT relation to the 2d Liouville theory, the Seiberg-Witten prepotential is constructed from the Bohr-Sommerfeld periods of 1d sine-Gordon model. If the same construction is literally applied to monodromies of exact wave functions, the prepotential turns into the one-parametric Nekrasov prepotential F(a,\epsilon_1) with the other epsilon parameter vanishing, \epsilon_2=0, and \epsilon_1 playing the role of the Planck constant in the sine-Gordon Shroedinger equation, \hbar=\epsilon_1. This seems to be in accordance with the recent claim in arXiv:0908.4052 and poses a problem of describing the full Nekrasov function as a seemingly straightforward double-parametric quantization of sine-Gordon model. This also provides a new link between the Liouville and sine-Gordon theories.